>A PROOF of the axioms of ZFC from FOL would be something indeed.
>The reason the Logicist programme (fails, is failing, may fail) is
>that in order to strengthen FOL to give you "a proof of the axioms of
>ZFC" you end up with some equally strong (and from a certain point of
>view, equally odoriferous).
>Robbie Lindauer
I think you are stealing some bases here. This thread is called "how
much of math is logic" -- nowhere did I attempt to defend the
proposition that ALL of math is logic, let alone the even stronger
proposition that ZFC can be derived from "FOL" (by which I assume you
mean the modern first-order predicate calculus, which is not identical
with "logic").
What I *am* proposing, first of all, is that the finite part of
mathematics (equivalently, Peano Arithmetic, or the theory of
hereditarily finite sets, etc.) can be derived from axioms which are
"logical" in character. To put it another way, I am trying to argue
that the non-logical content of mathematics comes essentially from the
treatment of infinite completed entities -- without an axiom of
Infinity (in Raatikainen's second, stronger sense), one cannot get
beyond PA, but that's still enough for an awful lot of math.
This argument requires "logic" to be more than simply the modern
first-order predicate calculus, but not much more. All you need is a
very weak theory of classes, or alternatively some very weak
first-order axioms in the language of sets (Friedman has investigated
how weak you can get away with, you need that the empty set exists and
that you can add individual members to sets to get new sets). Something
like this has been done by Frege, Russell, and others, in a strict
deductive way.
The real reason logicism became passe was that the limits of finite
mathematics became clear; which is why I am also asking for comment on
the proposition, not that "math is logic", but that "math is logic PLUS
the axiom of infinity". (Set theorists who go beyond ZFC and take
offense at the word "the" may address the modified proposition "math is
logic plus axioms of infinity".)
My second proposal is based on the observation a that stronger form of
second-order logic allows one to express the Continuum Hypothesis, so
that determining logical validity in this strong setup is at least as
hard as answering CH (conversely, if CH is actually indeterminate then
this form of "logic" is illegitimate because there is a sentence whose
validity status is undefined). By drawing a distinction between
deductive calculus and semantics, I can say that as far as semantics is
concerned, even more of math is logic. To say that CH is a logical
question does not make it any easier to answer, but it does address
philosophical concerns about the meaningfulness of CH. I therefore
asked whether there is any question in ordinary mathematics that would
not be "settled" by an oracle for "second-order validity" in the
standard semantics for second-order logic.
This is logicism in a different sense, because we no longer have a
complete deductive calculus. The point here is to address philosophical
questions about meaningfulness -- whether mathematics requires an
external "subject matter" or whether it is already implicit in a pure
"theory of concepts".
I will now try to ask some more precisely focused questions:
1) Do objections to set theory as "non-logical" even apply to the
extremely weak axioms for sets needed to develop the usual theory of
hereditarily finite sets, which is naturally bi-interpretable with PA?
2) If the answer to 1) is positive, is there any way whatsoever in
which we can talk about "classes" or "concepts" that deserves to be
called "logical"? (Because the properties of classes or concepts needed
to build up something equivalent to PA are also very weak.)
3) Why might ZFC be regarded as "odoriferous"?
4) If second-order logic is really "set theory in disguise", exactly
how much set theory does it disguise? Has anyone ever claimed that all
of ZFC could be recovered from a deductive calculus for second-order
logic?
-- JS
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